Theorem sbcbi2 2887

Description: Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Assertion

Ref	Expression
sbcbi2	|- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Proof of Theorem sbcbi2

Step	Hyp	Ref	Expression
1		abbi 2201	. . 3 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )
2		eleq2 2151	. . 3 |- ( { x | ph } = { x | ps } -> ( A e. { x | ph } <-> A e. { x | ps } ) )
3	1, 2	sylbi 119	. 2 |- ( A. x ( ph <-> ps ) -> ( A e. { x | ph } <-> A e. { x | ps } ) )
4		df-sbc 2839	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
5		df-sbc 2839	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6	3, 4, 5	3bitr4g 221	1 |- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1287   = wceq 1289   e. wcel 1438   {cab 2074   [.wsbc 2838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2839

This theorem is referenced by: (None)